is required. For the case \(F=0\) , the authors in [1]} highlighted the difficulty in obtaining a comprehensive singularity expansion of \(G(x,z)\mid _{x,z\in \partial M, x\ne z}\)  in a neighbourhood of the diagonal \(\lbrace x=z\rbrace \)  when \(M\)  is a bounded domain in \(\mathbb {R}^n\) , but it turns out that even when \(M\)  is a general Riemannian manifold this question can be treated via pseudo-differential techniques, which is done in [2]}. Here, we generalize result of [2]} for the case \(F\ne 0\) . Knowledge of the singularity structure allows us to derive the mean first arrival time of a Brownian particle on a Riemannian manifold with a single absorbing window which is a small geodesic ellipse. Our method extends to multiple windows but we present the single window case to simplify notations.
